Here is what the scholar Maynard Solomon has to say about the spelling of Mozart’s middle name (taken from Mozart: A Life, published in 1995, p. 277):
The reader of Mozart’s letters must soon grow accustomed to the numerous permutations to which the composer, who was baptized Joannes Chrysostomus Wolfgangus Theophilus Mozart, subjected his own name, apparently just for the fun of it. Thus, we are not surprised to find such variations on the surname as “De Mozartini,” “Mozartus,” or “Mozarty,” though we may be momentarily startled by such anagrammatical variants as “Trazom” or “Romzntz.” The name Wolfgang, though occasionally “Gangflow,” underwent comparatively few transformations. It is the last of Mozart’s forenames, Theophilus, that was subjected to the greatest variety of metamorphoses. Writing to his Augsburg publisher shortly after the birth, on 27 January 1756, of his son, Leopold Mozart noted, “The boy is called Joannes Chrisostomos, Wolfgang, Gottlieb,” thus translating the Greek Theophilus into the German Gottlieb, a form that was preserved – as “J.G. Wolfgang Mozart” – on the earliest of the child’s published works to bear opus numbers. From about 1770 on, Mozart several times referred to himself as “Wolfgango Amadeo Mozart,” but by 1778 he had adopted his favorite – and almost invariable – form of the name, Amadè or Amadé, with occasional ventures into Amadi or the Latinized Amadeus, the latter having originated as early as 1774 in a typically jesting message to his sister. (The almost universal adoption of the form Amadeus is a posthumous process, propelled in large part, I believe, by the wide circulation of Breitkopf & Härtel’s Oeuvres complètes de Wolfgang Amadeus Mozart (1798-1806). [And, I (Robert Markow) might add, by Milos Forman’s film Amadeus.]
Also, note that the famous signature reproduced in so many books, pamphlets, recordings, program notes, and other documents, is clearly Amadè, with the accent grave.